, Volume 190, Issue 14, pp 2677–2716

The Church–Fitch knowability paradox in the light of structural proof theory



Anti-realist epistemic conceptions of truth imply what is called the knowability principle: All truths are possibly known. The principle can be formalized in a bimodal propositional logic, with an alethic modality \({\diamondsuit}\) and an epistemic modality \({\mathcal{K}}\) , by the axiom scheme \({A \supset \diamondsuit \mathcal{K} A}\) (KP). The use of classical logic and minimal assumptions about the two modalities lead to the paradoxical conclusion that all truths are known, \({A \supset \mathcal{K} A}\) (OP). A Gentzen-style reconstruction of the Church–Fitch paradox is presented following a labelled approach to sequent calculi. First, a cut-free system for classical (resp. intuitionistic) bimodal logic is introduced as the logical basis for the Church–Fitch paradox and the relationships between \({\mathcal {K}}\) and \({\diamondsuit}\) are taken into account. Afterwards, by exploiting the structural properties of the system, in particular cut elimination, the semantic frame conditions that correspond to KP are determined and added in the form of a block of nonlogical inference rules. Within this new system for classical and intuitionistic “knowability logic”, it is possible to give a satisfactory cut-free reconstruction of the Church–Fitch derivation and to confirm that OP is only classically derivable, but neither intuitionistically derivable nor intuitionistically admissible. Finally, it is shown that in classical knowability logic, the Church–Fitch derivation is nothing else but a fallacy and does not represent a real threat for anti-realism.


Church–Fitch’s paradox Knowability principle Structural proof theory Proof analysis Intuitionistic bimodal logic Labelled sequent calculus 


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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity of FlorenceFlorenceItaly
  2. 2.Department of PhilosophyUniversity of Paris 1 Panthéon-SorbonneParisFrance
  3. 3.Department of PhilosophyUniversity of HelsinkiHelsinkiFinland

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